সংখ্যার অঙ্ক

For discussing Olympiad Level Number Theory problems
shehab ahmed
Posts:66
Joined:Tue Mar 20, 2012 12:52 am
সংখ্যার অঙ্ক

Unread post by shehab ahmed » Sat Apr 07, 2012 2:15 pm

Find all positive integers $n$ the product of whose digits is equal to $n^2 - 10n - 22$
Last edited by *Mahi* on Sat Apr 07, 2012 3:06 pm, edited 1 time in total.
Reason: LaTeXed

shehab ahmed
Posts:66
Joined:Tue Mar 20, 2012 12:52 am

Re: সংখ্যার অঙ্ক

Unread post by shehab ahmed » Sat Apr 07, 2012 8:03 pm

Let the product of the digits of $n$ be $P(n)$.
It is easy to prove that
$P(n)\leq n$.
So write $P(n)=n - k$
where $k \in \mathbb {N}$.
So we get the quadratic equation
$n^2 - 10n - 22=n - k$
the determinant of which must be a perfect square.Plugging the convenient values of $k$ and examining the related value of $n$,we get the only number that satisfies the condition is $12$
Last edited by sourav das on Sat Apr 07, 2012 8:33 pm, edited 1 time in total.
Reason: Latexed and edited

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